Logarithmic Bundles of Multi-Degree Arrangements in P^n

Let $ D = {D_{1}, \ldots, D_{\ell}} $ be a multi-degree arrangement with normal crossings on the complex projective space $ \{P}^n $, with degrees $ d_{1}, \ldots, d_{\ell} $; let $ \Omega_{\{P}^n}^1(\log D) $ be the logarithmic bundle attached to it. First we prove a Torelli type theorem when $ D $ has a sufficiently large number of components by recovering them as unstable smooth irreducible degree-$ d_{i} $ hypersurfaces of $ \Omega_{\{P}^n}^1(\log D) $. Then, when $ n = 2 $, by describing the moduli spaces containing $ \Omega_{\{P}^2}^1(\log D) $, we show that arrangements of a line and a conic, or of two lines and a conic, are not Torelli. Moreover we prove that the logarithmic bundle of three lines and a conic is related with the one of a cubic. Finally we analyze the conic-case.

2010 Mathematics Subject Classification: 14J60, 14F05, 14C34, 14C20, 14N05

Keywords and Phrases: Multi-degree arrangement, Hyperplane arrangement, Logarithmic bundle, Torelli theorem

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